Quantum tricritical point in the temperature-pressure-magnetic field phase diagramof CeTiGe3

Udhara S. Kaluarachchi,1, 2 Valentin Taufour∗,1 Sergey L. Bud’ko,1, 2 and Paul C. Canfield1, 2

1The Ames Laboratory, US Department of Energy, Iowa State University, Ames, Iowa 50011, USA2Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, U.S.A.

(Dated: July 12, 2018)

We report the temperature-pressure-magnetic field phase diagram of the ferromagnetic Kondo-lattice CeTiGe3 determined by means of electrical resistivity measurements. Measurements upto ∼ 5.8 GPa reveal a rich phase diagram with multiple phase transitions. At ambient pressure,CeTiGe3 orders ferromagnetically at TC = 14 K. Application of pressure suppresses TC, but a pres-sure induced ferromagnetic quantum criticality is avoided by the appearance of two new successivetransitions for p> 4.1 GPa that are probably antiferromagnetic in nature. These two transitions aresuppressed under pressure, with the lower temperature phase being fully suppressed above 5.3 GPa.The critical pressures for the presumed quantum phase transitions are p1∼= 4.1 GPa and p2∼= 5.3 GPa.Above 4.1 GPa, application of magnetic field shows a tricritical point evolving into a wing structurephase with a quantum tricritical point at 2.8 T at 5.4 GPa, where the first order antiferromagnetic-ferromagnetic transition changes into the second order antiferromagnetic-ferromagnetic transition.

INTRODUCTION

Quantum phase transitions (QPT) in metallic ferro-magnets have been studied for many years and remaina subject of great current interest [1]. The paramagnetic(PM) to ferromagnetic (FM) transition can be suppressedwith nonthermal control parameters such as pressure,chemical composition or external field often leading toa T = 0 K, QPT. However, according to the current the-oretical models, when suppressing the FM phase with aclean parameter such as pressure, a continuous PM toFM transition is not possible. Instead, the transition be-comes of the first order or a modulated magnetic phasecan appear. The possibility of a first-order transition orthe appearance of modulated magnetic phases was firstdiscussed in Ref. 2 and 3. In the case of the transition be-coming of the first order, a wing structure was predictedin Ref. 4 and observed in UGe2 [5] and ZrZn2 [6]. Thecase of the appearance of a modulated magnetic phaseis more complex[2, 3, 7–12] and an experimental exam-ples were found in LaCrGe3 [12] and CeRuPO [13]. Ob-servation of both tricritical wings and modulated mag-netic phase in LaCrGe3 is a good example of a complexphase diagram and provides a new example of the rich-ness of the phase diagram of metallic quantum ferromag-nets [14]. Recently, Belitz and Kirkpatrick proposed thatsuch complex phase diagram is due to quantum fluctua-tion effects [15].

Cerium based compounds have attracted attentiondue to interesting ground states, such as heavy-fermion,unconventional superconductor [16, 17], Kondo insu-lator [18], magnetic ordering [19, 20], etc. Whereasmany Ce-based compounds manifest an antiferromag-netic (AFM) ground state, only few systems areknown with FM order and pronounced Kondo ef-fects. CeRuPO [13], CeAgSb2 [21, 22], CeNiSb3 [23],CePd2Ge3 [24] and Ce2Ni5C3 [25] are some examples of

the Ce-based ferromagnets, which show complex phasediagrams under the application of pressure. Interestingly,the FM transition in these materials is suppressed withthe pressure and new magnetic (most probably AFM)phases appear before the Curie temperature reaches 0 Kbut no wing structure in the T -H-p phase diagrams hasbeen observed so far. According to the recent theoret-ical work by Belitz et al. [15], it is possible to have un-observable tricritical wings inside the AFM dome. Inmost of these cases, lack of in-field measurements un-der pressure prevents from constructing the temperature-pressure-field phase diagram and getting a better under-standing of the system. Therefore, it is interesting tofurther investigate the temperature-pressure-field effecton a Ce-based ferromagnetic system. To address this,we present measurements of electrical resistivity underpressure up to ∼ 5.8 GPa and magnetic field up to 9 T onferromagnetic CeTiGe3.

CeTiGe3 is one of the relatively rare examples of aferromagnetic Kondo lattice (γ=75 mJ mol1 K2 [26]); itorders with a Curie temperature, TC = 14 K [27]. It crys-tallizes in the hexagonal perovskite BaNiO3 - type struc-ture (P63/mmc) [27]. Magnetization measurements showhighly anisotropic behavior with c-axis being the easyaxis of magnetization [26]. A Curie-Weiss fit to the sus-ceptibility data yields an effective moment of 2.5µB, con-sistent with the reported values [26] and nearly equal tothe value for free-ions trivalent Ce (2.54µB). The re-ported saturation moment at 2 K from the magnetiza-tion data (1.72µB/Ce) along the c-axis [26] is compa-rable with the value obtained from the neutron diffrac-tion study(1.5µB/Ce) [28]. Substitution of titanium byvanadium (CeTi1−xVxGe3) causes a suppression of theCurie temperature down to 3 K at x= 0.3 and suggests apossible quantum critical point or phase transition nearx ≈ 0.35 [28]. In contrast to the effect of substitu-tion, a very small, initial positive pressure derivative of

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TC (dTC/dp≈ 0.3 K GPa−1 up to 1 GPa) suggests thatCeTiGe3 is located near the maximum of the magneticordering temperature in the Doniach model [28]. How-ever, all substitution and pressure measurements havebeen done on the polycrystalline material and only tomodest pressure, p< 1 GPa. To get a better understand-ing of T -p-H phase diagram, possible FM instability andQCP it is important to perform high pressure studies onsingle crystalline samples of CeTiGe3 over a wide pres-sure range.

EXPERIMENTAL METHODS

Single crystals of CeTiGe3 were grown using a hightemperature solution growth technique [29, 30]. A mix-ture of elemental Ce, Ti and Ge was placed in a 2mL fritted alumina crucible [31] with a molar ratio ofCe:Ti:Ge = 4:1:19 [26] and sealed in a silica ampule undera partial pressure of high purity argon gas. The sealedampule was heated to 1200 ℃ over 10 hours and heldthere for 5 hours. It was cooled to 900℃ over 120 hoursand excess liquid was decanted using a centrifuge. Agood quality sample (based on the residual resistivity ra-tio) for the pressure study was selected after ambientpressure characterization by the magnetization and re-sistivity measurements. Temperature and field depen-dent resistance measurements were carried out using aQuantum Design (QD) Physical Property MeasurementSystem (PPMS) from 1.8 K to 300 K. The ac-resistivity(f = 17 Hz) was measured by the standard four-probemethod with the 1 mA current in the ab plane. FourAu wires with diameters of 12.5µm were spot welded tothe sample. A magnetic field, up to 9 T, was appliedalong the c-axis, which corresponds to the magnetizationeasy axis [26]. A modified Bridgman cell [32] was used togenerate pressure for the resistivity measurement. A 1:1mixture of n-pentane:iso-pentane was used as a pressuremedium. The solidification of this medium occurs around∼6-7 GPa at room temperature [33–37]. The pressure atlow temperature was determined by the superconductingtransition temperature of Pb [38].

RESULTS AND DISCUSSION

The temperature dependencies of the in-plane resistiv-ity of single crystalline CeTiGe3 under various pressuresup to 5.76 GPa are shown in Fig. 1 (a). At ambient pres-sure, the resistivity exhibits typical Kondo-lattice behav-ior with a broad minimum ∼ 190 K followed by a maxi-mum at Tmax = 31 K. The Tmax is assumed to be relatedto the Kondo interaction with a changing population ofcrystal electric field levels [26, 39–41]. The FM transitionmanifests itself in the resistivity data as a sharp dropat TC = 14.2 K. Similar values of TC have been reported

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FIG. 1. (Color online) (a) Temperature dependence of thein-plane resistivity, ρ(T ), of a CeTiGe3 single crystal undervarious pressures, p, up to 5.76 GPa on a semi-log plot. Theresistivity at 300 K linearly increase with the pressure at arate of 7.4µΩ cm GPa−1 from 0 to 5.76 GPa as shown in theinset. (b) Low temperature resistivity at various pressures.Data are offset by increments of 10µΩ cm for clarity.

from polycrystalline and single crystalline samples [26–28]. The residual resistivity ratio (RRR) is 19, a valuethat suggests a rather good quality of the sample. Uponapplication of pressure the resistivity at room temper-ature increases linearly with a rate of 7.4µΩ cm GPa−1

over the whole pressure range (see inset of Fig. 1 (a)),both the local maximum and local minimum in the re-sistivity broaden and move to higher temperatures withincreasing pressure. The evolution of the low tempera-ture resistivity is shown in Fig. 1 (b); data are offset byincrements of 10µΩ cm for clarity.

Figure 2 shows the evolution of the low temper-ature resistivity and its temperature derivatives inthree selected pressure regions; (I) p< 4.1 GPa (II)4.1 GPa 5.3 GPa. Below

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FIG. 2. (Color online) Low temperature, in-plane resistiv-ity (left axis) and its corresponding temperature derivative(right axis) of CeTiGe3 for several representative pressure re-gions (a)-(b) p< 4.1 GPa, (c)-(e) 4.1 GPa

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FIG. 3. (Color online) Evolution of the temperature deriva-tive of the resistivity at low temperature for representativepressures. The data are vertically offset by 28µΩ cm K−1 toreduce overlap. Solid symbols represent the criteria describedin Fig.2. At 5.29 GPa there is an additional anomaly in thedρ/dT as shown by the orange circle.

sults show breaks in ρ1.8K(p) at p1 (FM to MP2) anda maximum at p2 (MP2 to MP1). The exact nature ofthe phase transitions at p1 and p2 are not known andto resolve this, it would be useful to study the magneticordering wave vector under pressure.

Application of an external magnetic field adds anotherdimension to our phase diagram and different behaviorof the resistivity anomalies under magnetic field allowus to explore further new phase regions of this mate-rial. Figure 5 (a) shows the temperature dependence of ρat different magnetic fields, applied along the c-axis, at4.48 GPa. The sharp drop in the resistivity at low fields(µ0H ≤ 0.3 T) broadens at higher fields. These data man-ifest hysteretic behavior up to 0.5 T, indicating the firstorder nature of the transition. The zero-field kink inthe resistivity, at 9.8 K, changes into a hump with theincrease of field (0.25 T) and disappears at 0.3 T. An-other hump like feature appears above 0.35 T and broad-ens with further increase of the field. These features canbe clearly observed in temperature derivative shown inFig. 5 (b).

The field dependence of ρ (p= 4.48 GPa) below 7 Kshows a metamagnetic transition with a low field plateaufollowed by a step-like feature and develops into two

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FIG. 4. (Color online)(a) T − p phase diagram of CeTiGe3in zero applied field. Transition temperatures are determinedfrom the anomalies in dρ/dT as shown in Fig. 2 and Fig. 3 .The values of critical pressure p1 and p2 are 4.1 and 5.3 GParespectively. Solid lines are guide to the eye and dashedlines are suggested extrapolations of phase boundaries. Thered and blue color line represent the second and first orderphase transitions. (b) Maximum in resistivity, Tmax (shownin Fig. 1(a)), as a function of pressure. (c) Pressure depen-dence of the ρ at 1.8 K.

transitions above 7 K (Fig. 5 (c)). The solid and dashedlines represent the field increasing (ρup(H)) and de-creasing (ρdown(H)) respectively. The difference be-tween ρup(H)-ρdown(H) shows a sizable deviation (ρ issmaller in the increasing-field than the decreasing-field)for 0≤H ≤ 0.3 T range. In Fig. 5, hysteresis is apparentnot only in the transition temperature (Fig. 5 (a)) andtransition field (Fig. 5 (c)), but also in the magnitude ofthe resistivity. Similar hysteretic behavior is observedin the CeAuSb2 [43–45] and CeTAl4Si2 (T = Rh, Ir) [46]. Based on the hysteretic behavior, we can concludethese metamagnetic transitions are likely associated witha first-order phase transition. The observed hysteresis inthe magnitude of resistivity indicates the possibility ofmagnetic domains. At temperatures above 11 K, the re-sistivity shows a very broad anomaly and no transitionhas been observed. Criteria used to obtain transitionfields are shown in the inset of Fig. 5 (c).

Figures 6 (a)-(d) show the T − H phase diagrams at

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FIG. 5. (Color online)(a) Temperature dependence of theresistivity at various fixed fields for p= 4.48 GPa and H || c.The data are vertically shifted by integer of 25µΩ cm to avoidoverlapping. The insets show the observed hysteretic behaviorin temperature scan. Continuous and dashed lines representthe temperature increasing and decreasing respectively. (b)Corresponding temperature derivative (dρ/dT ) of (a). Thedata are vertically shifted by integer of 15µΩ cm K−1 to avoidoverlapping. Solid symbols represent the criteria use to ob-tain the teansition temperatures at various magnetic fields.(c) Field dependence of the resistivity at fixed temperatures.For these data the sample was cooled in zero field and thenρ(H) data was collected for increasing field (ρup) and thendecreasing field (ρdown). Then increase the temperature tothe desired value and data was collected for increasing anddecreasing field. Continuous and dashed lines represent thefield increasing and decreasing respectively. Insets show theobserved hysteretic behavior and the criteria used to obtainthe transition fields. Above 7 K no hysteretic behavior is ob-served.

representative pressures. Transition temperatures deter-

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FIG. 6. (Color online) T − H phase diagrams for variouspressures: (a) 4.48 GPa (b) 4.86 GPa (c) 5.15 GPa and (d)5.46 GPa, determine by tracking various anomalies in tem-perature and field derivatives of resistivity measurement asshown by Fig. 5. Solid and open symbols represent transitiontemperatures determined by T -sweeps and transition fieldsdetermined by H-sweeps (as described in Fig. 5) respectively.Continuous blue and red lines indicate the first order and sec-ond order transitions respectively. See appendix for T − Hphase diagrams for all the measured pressures. (e) Tempera-ture dependence of hysteresis widths for the transition at H1at various pressures. The data are vertically offset by 0.03 Tto avoid overlap. Vertical arrows represent the estimated tri-critical points for each pressure. Zero for each data set shownon right-hand axis.

mined by T -sweep measurements are shown by closedsymbols and anomalies appeared in isothermal H-sweepmeasurements are shown by open symbols. Continuousblue and red lines indicate the first order and second or-der transitions respectively (based on the presence or lackof hysteretic behavior respectively). The red circle repre-sents the tricritical point (TCP) determined by Fig. 6 (e).

6

Temperature dependence hysteresis widths for the transi-tion at H1 are shown in Fig. 6 (e). The data are verticallyoffset by 0.03 T to avoid overlap. Clear hysteresis at lowtemperature gradually decreases with increasing temper-ature and disappears at a TCP as shown by a verticalarrow. In contrast to the wing-critical-point (WCP) inUGe2 [5] and LaCrGe3 [14], here we observed a TCP inthe T − H phase diagram where first order transitionchanges into the second order transition. This TCP cor-responds to the boundary of the wing structure similar toUGe2 [5] and LaCrGe3 [14]. The T−H phase diagrams ofCeTiGe3 for pressures between 4.1 - 5.3 GPa show com-plex behavior. Three magnetic phases (MP1, MP1' andMP2) are identified by the anomalies in the resistivitymeasurement. Both MP1 and MP1' phases are separatedby MP2 phase by a first order transition as shown inFigs. 6 (a)-(c). For pressures between 4.1-5.3 GPa, theseT − H phase diagrams are similar to those found forCeRu2Al2B [47], which undergoes a second order AFMtransition that is followed by a first order FM transitionas a function of temperature. Above 5.3 GPa, only twomagnetic phases; MP1' and MP4 are identified by theresistivity measurements and there is no longer a firstorder phase transition boundary observed. The T − Hphase diagrams for all the pressures above 4.13 GPa areshown in Fig. 10.

Figure 7 (a) shows the field dependence of the resistiv-ity at 1.8 K, ρ(H), for different pressures. For the pres-sures in between p1 and p2, ρ(H) for an increasing mag-netic field shows a clear metamagnetic transition with asubstantial (> 40%), drop of resistivity. For higher pres-sures, the sharp drop in the ρ(H) disappears and severalmetamagnetic transitions can be observed. Figures 7 (b)and (c) show the representative magnetoresistance datafor 4.1 GPa 5.3 GPa respectively.Transition fields determined by H-sweeps measurementsare shown by the open symbols. To estimate the tran-sition width, we used the field derivative of the resistiv-ity at 1.8 K, as shown in Figs. 7(d) and (e). The min-imum at H1 is fitted with Gaussian+linear-backgroundand obtained the width of the Gaussian distribution. Theblue color lines in Figs. 7(b) and (c) represent the fit-ted curves to the data. We noticed that the transitionwidth (Fig. 7 (f) right axis) at H1 at 1.8 K remains smallfor the first-order transition and becomes broad in thesecond-order regime. Using linear extrapolation as rep-resented by red dashed lines, we obtained pressure cor-responding to the TCP at 1.8 K, which is 5.3 GPa. Inaddition to that, the temperature dependence hysteresiswidth for transition H1 at 1.8 K is also suppressed withthe pressure and disappeared above 5.3 GPa as shown inFig. 7 (f) left axis. Figure 7 (g) shows the H − p phasediagram at 1.8 K constructed from the above criteria.The magnetic field that corresponds to the H1 transi-tion is shifted up with pressure. Its extrapolation downto zero yields p∼= 4.1 GPa, which is in agreement with

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FIG. 7. (Color online) (a) Field dependence of ρ at 1.8 Kfor various pressures. Continuous and dashed lines representthe field increasing and decreasing respectively. Representa-tive ρ(H) data for (b) 4.1 GPa 5.3 GPaand the criteria used to obtain the transition fields at 1.8 K.The open symbols represent the corresponding transitionfields. The gray star represents the shoulder like anomalyappeared at 5.76 GPa (see Appendix for more details). (d)-(e) Representative derivative, dρ/dH data for H1 transitionat 1.8 K. The blue color lines represent the Gaussian+linear-background fitted curves which are used to obtained full-width of the H1 transition. (f) left axis shows the pressuredependence hysteresis width of transition H1 at 1.8 K. Rightaxis shows pressure dependence of the full-width of H1 ob-tained by dρ/dH ((d)-(e)) at 1.8 K. Vertical dashed line rep-resent the tricritical pressure∼ 5.3 GPa, at 1.8 K. (g) H − pphase diagram at 1.8 K based on the criterion shown in (b,c).Blue and red solid lines represent the first and second or-der transitions. Red open circle represents the extrapolatedQTCP.

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the p1 obtained from T -p diagram (Fig. 4 (a)). We ob-serve the increasing rate of metamagnetic transition fieldwith respect to pressure, changes near 5.3 GPa. Simi-lar H − p phase diagrams at low temperature have beenobserved in LaCrGe3 [14] and CeRu2(Si1−xGex)2 sys-tem [48, 49]. CeRu2Ge2 is a local moment system [50],while CeRu2Si2 is itinerant [51]. Application of pressureto CeRu2Ge2 gives nearly same magnetic phase diagramas that of CeRu2(Si1−xGex)2 [52, 53]. Observed trans-port and de Haas-van Alphen data suggest that, for thissystem, change of the f - electron nature from local toitinerant occurs when the FM phase disappears [48]. Onthe other hand, itinerant ferromagnet LaCrGe3 show tri-critical wings as well as modulated magnetic phase. In-terestingly, T−p−H phase diagram of both LaCrGe3 [14]and CeRu2Ge2 [49] without AFM states is similar to theitinerant weak ferromagnet like UGe2 [5]. This similaritymight imply that the physics behind these phase dia-grams are not very different.

The projection of the wing lines in T − H, T − pand H − p planes are shown in Figs. 8 (a),(b) and (c)respectively. The wing lines can be extrapolated to aquantum-tri-critical-point (QTCP) at 0 K, which is foundto be at 2.8 T at 5.4 GPa. Theoretical analysis basedon Landau expansion shows that the slope of the wingsdT/dH and dp/dH are infinite near H = 0 T [54]. Thiswas observed experimentally in URhGe [55]. It was alsoobserved in LaCrGe3, despite the existence of anothermagnetic phase [14]. Here, we do not observe such be-havior which could be due to the existence of the mag-netic phase MP1 or to the lack of data near p1. Morecareful measurements near p1. are required. Also, theTCP at H = 0 T is found to be ∼ 8 K and this is belowthe MP1 transition. A similar observation was made inLaCrGe3 [14] where the TCP seems to be located belowthe Lifshitz point. Recent theoretical description by Be-litz and Kirkpatrick in Ref. 15 shows the complex behav-ior of the phase diagrams of metallic magnets when anAFM order is observed in addition to the FM phase dueto the quantum fluctuations. Similar to the Fig. 4 (a) inRef. 15, we observed a QTCP where first order AFM-FMtransition changes into the second order AFM-FM tran-sition at 2.8 T at 5.4 GPa (see Fig. 7 (g)). Very recentlyQTCP has experimentally observed in NbFe2 [56].

The constructed, partial, T − p−H phase diagram ofCeTiGe3 based on resistivity measurements is shown inFig. 9. A FM QCP in CeTiGe3 is avoided by the appear-ance of MP1 and MP2 phases, and shows field inducedwing structure above 4.1 GPa. The estimated QTCP isshown by the open red circle. In order to provide clearpicture of the wing structure phase diagram, we showonly selected phases here (see Fig. 10 for H-T phase dia-grams at various pressures). In the case of the itinerantthe ferromagnet, LaCrGe3 [12, 14], the second-order FMtransition becomes a first order at a tricritical point inthe T -p plane and application of a magnetic field reveals

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FIG. 8. (Color online) Projection of wings in (a) T −H (b)T − p and (c) H − p planes. Red solid circles represent theTCP determined by Fig. 6(e). Red solid squared obtainedfrom Fig. 7 (f). Dashed lines are guides to the eyes and openred circles represent the extrapolated QTCP.

FIG. 9. (Color online) The constructed, partial, T − p − Hphase diagram of CeTiGe3 based on resistivity measurements.Blue color surfaces represent the first-order planes and greencolor surface represents the first-order MP2 phase boundary.Continuous red and blue lines represent the second and firstorder transition respectively. The open circle represent ex-trapolated QTCP.

a wing structure phase diagram. Appearances of modu-lated magnetic phase in LaCrGe3 [14] makes it the firstexample of new type of phase diagram of metallic quan-tum ferromagnets. Unlike LaCrGe3 (Fig. 5 in Ref. [14]),where, wings are extended beyond the AFM phases, theobserved wings in CeTiGe3 are always bounded by theAFM phases. This can be clearly visualized in Fig. 7 (g)(for comparison see Fig. 4 in Ref. [14]). The observationof QTCP in metallic magnets in the case of appearanceof AFM order in addition to the FM order is theoreticallydescribed by Belitz and Kirkpatrick [15]. This theoretical

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finding is consistent with our experimental observation ofQTCP in CeTiGe3. Therefore, CeTiGe3 is a good exam-ple of a Ce-based compounds in which the system can bedriven into various magnetic ground state by fine tun-ing of the exchange interaction achieved by temperature,pressure and magnetic field.

CONCLUSIONS

We have measured the high pressure electrical resis-tivity of CeTiGe3 up to 5.8 GPa and 9 T and found acomplex T − p − H phase diagram. The ferromagnetictransition at ambient pressure initially slightly increasesand then decreases, indicates that CeTiGe3 is locatedjust below the maximum (left side) of the Doniach phasediagram. The ferromagnetic transition suppresses near4.1 GPa and cascade of phase transitions are observedabove that. Change in residual resistivity near 4.1 GPaand 5.3 GPa suggests a modification of the electronicstructure upon entering these magnetic phases. Thus,CeTiGe3 is another clear example of avoided ferromag-netic quantum critical point due to appearance of mag-netic phase (probably antiferromagnetic). Application ofmagnetic field under pressure above 4.1 GPa reveals wingstructure phase diagram. In contrast to the wing criti-cal point in LaCrGe3, we observed a tricritical point inH-p plane, which corresponding to the boundary of thewing structure. Estimated quantum tricritical point ofCeTiGe3 is located at 2.8 T at 5.4 GPa. We believe thatthe present work will stimulate further experiments toinvestigate the properties of this material.

ACKNOWLEDGMENTS

We would like to thank S. Manni and A. Kreyssig foruseful discussions. This work was supported by the U.S.Department of Energy (DOE), Office of Science, BasicEnergy Sciences, Materials Science and Engineering Di-vision. The research was performed at the Ames Labora-tory, which is operated for the U.S. DOE by Iowa StateUniversity under contract No. DE-AC02-07CH11358.V.T. was partially supported by Critical Material Insti-tute, an Energy Innovation Hub funded by U.S. DOE, Of-fice of Energy Efficiency and Renewal Energy, AdvancedManufacturing Office.

Appendix

Figure 10 shows the constructed T−H phase diagramsfor pressures between 4.21 to 5.76 GPa. There is a cleardifference in the T −H phase diagrams below 4.86 GPaand above 5.46 GPa. T −H phase diagram for the inter-mediate pressure, 5.29 GPa, shows a complex behavior.

Also, we observed an additional shoulder-like anomalyin ρ(H) at 5.76 GPa (gray color star in Fig. 7 (c) andFig. 10). When the temperature was increased, it becamebroadened and merged with H1 and no loner resolvable.H1, H2 and H3 are the anomalies observed in ρ(H) dataas shown in Fig. 5 (c) and Figs. 7 (b)-(c)

9

F M

0 . 2 0 . 4 0 . 6 0 . 8002468

1 01 2

0 . 2 0 . 4 0 . 6 0 . 80

M P 1 M P 3

0 . 2 0 . 4 0 . 6 0 . 80

M P 1 M P 3

M P 2

0 . 5 1 . 0 1 . 5002468

1 01 2

M P 3M P 1

M P 2

0 . 5 1 . 0 1 . 50

M P 1 M P 1 '

M P 2

0 . 5 1 . 0 1 . 50

M P 4

?

0 . 5 1 . 0 1 . 5 2 . 0 2 . 5002468

1 01 2

1 2 3 4 5 60

M P 4

1 2 3 4 5 60

d � / d T m i d d � / d T p e a k d � / d T k i n k 1 d � / d T k i n k 2T

(K)

µ0 H ( T )

4 . 2 1 G P a

M P 2

M P 1T C P

µ0 H ( T )

4 . 3 8 G P a

M P 2

M P 1 M P 3

µ0 H ( T )

4 . 4 8 G P a

M P 2

T (K)

µ0 H ( T )

4 . 7 2 G P a 4 . 8 6 G P a

µ0 H ( T ) µ0 H ( T )

5 . 1 5 G P a

H 2 [ k i n k 2 i n � ( H ) ] H 3 [ k i n k 3 i n � ( H ) ]

H 1 [ k i n k 1 i n � ( H ) ]

M P 2

5 . 2 9 G P a

T (K)

µ0 H ( T )

M P 1 '

µ0 H ( T )

5 . 4 6 G P a

M P 1 ' M P 4

µ0 H ( T )

5 . 7 6 G P a

M P 1 '

FIG. 10. (Color online) T −H phase diagrams, including those shown in Fig. 6 (a)-(d,) at various increasing applied pressures.At 5.29 GPa, T −H phase diagrams show a complex behavior and with additional metamagnetic transitions (gray and brownopen triangles) in ρ(H) data (raw data are not shown). H1, H2 and H3 are the anomalies observed in ρ(H) data as shown inFig. 5 (c) and Figs. 7 (b)-(c).

∗ Current affiliation: Department of Physics, Univer-sity of California, Davis, California 95616, USA.

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Quantum tricritical point in the temperature-pressure-magnetic field phase diagram of CeTiGe3Abstract Introduction Experimental Methods Results and Discussion Conclusions ACKNOWLEDGMENTS Appendix References